A woman 5 ft tall is standing near a street lamp that is 12 ft tall. Find a function that models the length L of her shadow in terms of her distance d from the base of the lamp.
This one's kind of pretty. I first thought we'd be into sin and trig country, but we don't have to. But you have to see it in a diagram.
Draw lines for the lamppost and the woman, and a line from the top of the lamppost just over the woman to the ground. You should have a big RAT, with the woman forming a similar, smaller RAT, included. Distance from lamppost to woman is d; length of shadow is s.
Key word is similar. The ratio of the sides is the same in the big and small triangles.
The base of the big triangle is s+d, and its height is 12.
The base of the small triangle is s, and its height is 5.
So we have:
(s+d)/12 = s/5
and now all you have to do is manhandle that equation to get s in terms of d!
S=5d/7
To find a function that models the length L of the woman's shadow in terms of her distance d from the base of the lamp, we can use similar triangles. Let's denote the length of the woman's shadow as x.
Since the woman and the street lamp are similar triangles, we can set up the proportion:
(Height of woman)/(Length of woman's shadow) = (Height of lamp)/(Length of lamp's shadow)
(5 ft) / x = (12 ft) / d
To find the function that models the length L of her shadow, we solve this proportion for x. First, cross multiply:
(5 ft) * (d) = (12 ft) * x
5d = 12x
Now, we can isolate x by dividing both sides by 12:
x = (5d) / 12
Therefore, the function that models the length L of her shadow in terms of her distance d from the base of the lamp is:
L(d) = (5d) / 12
To find a function that models the length of the woman's shadow in terms of her distance from the base of the lamp, you can use similar triangles.
Let's assume that the length of the woman's shadow is L and her distance from the base of the lamp is d. We can set up a proportion using the similar triangles:
Height of the woman / Length of her shadow = Height of the lamp / Distance from the base of the lamp
Since the height of the woman is 5 ft and the height of the lamp is 12 ft, we have:
5 ft / L = 12 ft / d
To solve for L, we can cross-multiply:
5d = 12L
Now, we can divide both sides of the equation by 5:
d = (12/5)L
This equation represents the relationship between the length of the woman's shadow (L) and her distance from the base of the lamp (d).